Differential Equations

Lecture 1: The Geometrical View of y’=f(x,y): Direction Fields, Integral Curves


Lecture 2: Euler’s Numerical Method for y’=f(x,y) and its Generalizations


Lecture 3: Solving First-order Linear ODE’s; Steady-state and Transient Solutions


Lecture 4: First-order Substitution Methods: Bernouilli and Homogeneous ODE’s


Lecture 5: First-order Autonomous ODE’s: Qualitative Methods, Applications


Lecture 6: Complex Numbers and Complex Exponentials


Lecture 7: First-order Linear with Constant Coefficients: Behavior of Solutions, Use of  Complex Methods


Lecture 8:​ Continuation; Applications to Temperature, Mixing, RC-circuit, Decay, and Growth Models


Lecture 9: Solving Second-order Linear ODE’s with Constant Coefficients: The Three Cases


Lecture 10: Continuation: Complex Characteristic Roots; Undamped and Damped Oscillations


Lecture 11: Theory of General Second-order Linear Homogeneous ODE’s: Superposition, Uniqueness, Wronskians


Lecture 12: Continuation: General Theory for Inhomogeneous ODE’s. Stability Criteria for the Constant-coefficient ODE’s


Lecture 13: Finding Particular Sto Inhomogeneous ODE’s: Operator and Solution Formulas Involving Exponentials


Lecture 14: Interpretation of the Exceptional Case: Resonance


Lecture 15: Introduction to Fourier Series; Basic Formulas for Period 2(pi)


Lecture 16: Continuation: More General Periods; Even and Odd Functions; Periodic Extension


Lecture 17: Finding Particular Solutions via Fourier Series; Resonant Terms; Hearing Musical Sounds


Lecture 19: Introduction to the Laplace Transform; Basic Formulas


Lecture 20: Derivative Formulas; Using the Laplace Transform to Solve Linear ODE’s


Lecture 21: Convolution Formula: Proof, Connection with Laplace Transform, Application to Physical Problems


Lecture 22: Using Laplace Transform to Solve ODE’s with Discontinuous Inputs


Lecture 23: Use with Impulse Inputs; Dirac Delta Function, Weight and Transfer Functions


Lecture 24: Introduction to First-order Systems of ODE’s; Solution by Elimination, Geometric Interpretation of a System


Lecture 25: Homogeneous Linear Systems with Constant Coefficients: Solution via Matrix Eigenvalues (Real and Distinct Case)


Lecture 26: Continuation: Repeated Real Eigenvalues, Complex Eigenvalues


Lecture 27: Sketching Solutions of 2×2 Homogeneous Linear System with Constant Coefficients


Lecture 28: Matrix Methods for Inhomogeneous Systems: Theory, Fundamental Matrix, Variation of Parameters


Lecture 29: Matrix Exponentials; Application to Solving Systems


Lecture 30: Decoupling Linear Systems with Constant Coefficients


Lecture 31: Non-linear Autonomous Systems: Finding the Critical Points and Sketching Trajectories; the Non-linear Pendulum


Lecture 32: Limit Cycles: Existence and Non-existence Criteria


Lecture 33: Relation Between Non-linear Systems and First-order ODE’s; Structural Stability of a System, Borderline Sketching Cases; Illustrations Using Volterra’s Equation and Principle

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